34. Wave Optics

Two satellites at an altitude of 1200 km are separated by 28 km. If they broadcast 3.6-cm microwaves, what minimum receiving-dish diameter is needed to resolve (by Rayleigh’s criterion) the two transmissions?

Your artist friend is designing an exhibit inspired by circular-aperture diffraction. A pinhole in a red zone is going to be illuminated with a red laser beam of wavelength 670 nm, while a pinhole in a violet zone is going to be illuminated with a violet laser beam of wavelength 410 nm. She wants all the diffraction patterns seen on a distant screen to have the same size. For this to work, what must be the ratio of the red pinhole’s diameter to that of the violet pinhole?

In the atom interferometer experiment of Figure 38.13, laser-cooling techniques were used to cool a dilute vapor of sodium atoms to a temperature of 0.0010 K=1.0 mK. The ultracold atoms passed through a series of collimating apertures to form the atomic beam you see entering the figure from the left. The standing light waves were created from a laser beam with a wavelength of 590 nm.

d. Because interference is observed between the two paths, each individual atom is apparently present at both point B and point C. Describe, in your own words, what this experiment tells you about the nature of matter.

The entrance to a boy’s bedroom consists of two doorways, each 1.0 m wide, which are separated by a distance of 3.0 m. The boy’s mother yells at him through the two doors as shown in Fig. 35–42, telling him to clean up his room. Her voice has a frequency of 400 Hz. Later, when the mother discovers the room is still a mess, the boy says he never heard her telling him to clean his room. The velocity of sound is 340 m/s . (a) Find all of the angles θ (Fig. 35–42) at which no sound will be heard within the bedroom when the mother yells. Assume sound is fully absorbed when it strikes a bedroom wall. (b) If the boy was at the position shown when his mother yelled, does he have a good explanation for not having heard her? Explain.

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(II) Monochromatic light falls on a transmission diffraction grating at an angle Φ to the normal. (a) Show that Eq. 35–13 for diffraction maxima must be replaced by

d(sinΦ + sinθ) = ±mλ. m = 0, 1, 2, ••• .

(b) Explain the ± sign. (c) Green light with a wavelength of 550 nm is incident at an angle of 15° to the normal on a diffraction grating with 4500 slits/cm . Find the angles at which the first-order maxima occur.

(II) A diffraction grating has 15,000 rulings in its 1.9 cm width. Determine (a) its resolving power in first and second orders, and (b) the minimum wavelength resolution (∆λ) it can yield for λ = 410 nm.

When yellow sodium light, λ = 589nm , falls on a diffraction grating, its first-order peak on a screen 62.0 cm away falls 3.32 cm from the central peak. Another source produces a line 3.71 cm from the central peak. What is its wavelength? How many slits/cm are on the grating?

A He–Ne gas laser which produces monochromatic light of wavelength λ = 6.328 x 10⁻⁷m is used to calibrate a reflection grating in a spectroscope. The first-order diffraction line is found at an angle of 18.5° to the incident beam. How many lines per meter are imprinted on this reflecting diffraction grating?

How many slits per centimeter must a grating have if there is to be no second-order spectrum for any visible wavelength?

(II) If X-ray diffraction peaks corresponding to the first three orders ( m = 1, 2, and 3) are measured, can both the X-ray wavelength λ and lattice spacing d be determined? Prove your answer.

(III) (a) Derive an expression for the intensity in the interference pattern for three equally spaced slits. Express in terms of δ = 2πd sin θ / λ where d is the distance between adjacent slits and assume the slit width D ≈ λ . (b) Show that there is only one secondary maximum between principal peaks.

(II) Suppose the angles measured in Problem 42 were produced when the spectrometer (but not the source) was submerged in water. What then would be the wavelengths (in air)?

(II) Red laser light from a He–Ne laser (λ = 632.8 nm) creates a second-order fringe at 53.2° after passing through a grating. What is the wavelength λ of light that creates a first-order fringe at 21.2°?

The wings of a certain beetle have a series of parallel lines across them. When normally incident 520-nm light is reflected from the wing, the wing appears bright when viewed at an angle of 56°. How far apart are the lines?

(a) How far away can a human eye distinguish two car headlights 2.0 m apart? Consider only diffraction effects and assume an eye pupil diameter of 6.0 mm and a wavelength of 560 nm. (b) What is the minimum angular separation an eye could resolve when viewing two stars, considering only diffraction effects? In reality, it is about of arc. Why is it not equal to your answer in (b)?

A beam of 125-eV electrons is scattered from a crystal, as in X-ray diffraction, and a first-order peak is observed at θ = 43°. What is the spacing between planes in the diffracting crystal? (See Section 35–11.)

What is the highest spectral order that can be seen if a grating with 6800 slits per cm is illuminated with 633-nm laser light? Assume normal incidence.

(III) Derive an expression for the intensity in the interference pattern for three equally spaced slits. Express in terms of δ = 2πd sin θ / λ where d is the distance between adjacent slits and assume the slit width D ≈ λ . (b) Show that there is only one secondary maximum between principal peaks.